## A Little Bit of Descartes
The 16th and 17th centuries hosted a proliferation of pre-scientific and scientific “Fathers of” figures: Galileo Galilei, the Father of Physics; Nicholas Copernicus and Johannes Kepler, arguably the Fathers of Astrophysics; and Tycho Brahe, the Father of Astronomy. That leaves out some lesser-known, but influential dads-of, like Roger Bacon, Frances Bacon (no relation), and Walter Gilbert, all of whom share paternity as Fathers of Experimentation. But the Granddaddy...um...Father of them all was René Descartes (1596-1650), often referred to as the Father of Western Philosophy and a Father of Mathematics, if not a favorite Uncle of Physics. If you’ve ever plotted a point in a coordinate system, you’ve paid homage to Descartes.
Frankly, if you’ve ever plotted a function, you’ve paid homage to Descartes. If you’ve ever looked at a rainbow? Yes. Him again. If you ever felt that the mind and the body are perhaps two different things, then you’re paying homage to Descartes and if you were taught to be skeptical of authority and to work things out for yourself? Descartes. But above all---for us---René Descartes was the Father of analytic geometry.
He was born in 1596 in a little French village called, Descartes---what are the odds? (Okay. That came later.) By this time Galileo was a professor in Padua inventing physics and Caravaggio was in Rome inventing the Baroque. Across the Channel Shakespeare was in London inventing theater and Elizabeth had cracked the Royal Glass Ceiling and was reinventing moderate political rule. This was a time of discovery when intellectuals began to think for themselves. This is the beginning of the end of the suffocating domination of Aristotle.
```{aside}
René Descartes.
1596-1650
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René was sent to a prominent Jesuit school at the age of 10 and a decade later emerged with his mandated law degree. Apart from his success in school, his most remarkable learned skill was his lifelong manner of studying: often ill, he was allowed to spend his mornings in bed, a habit he retained until the last year of his life. There’s a story there.
His school required physical fitness and in spite of his health, he became a proficient swordsman and soldier---wearing a sword throughout his life as befitting a “gentleman.” For a while he was essentially a soldier of fortune, alternating between raucous partying in Paris with friends and combat assignments (a Catholic, fighting with the Dutch Protestants) in various of the innumerable Thirty Years War armies.
Somewhere in that period Descartes became serious and decided that he had important things to say. He wrote a handful of unpublished tracts and became well-known through a steady correspondence with European intellectuals.
By 1628 he began to suspect that his ideas were not going to sit well in Catholic France (confirmed for him when Galileo was censured in 1633) and so he moved to Holland where he lived for more than 20 years. He'd been playing with mathematics during his playboy-soldier period and little did he know surprisingly, he found he was a mathematical genius, solving problems that others couldn't. He enrolled himself as a "mature student" in Leiden and devoted himself to mathematics. By 1637, he changed the landscape forever.
### Descartes’ Algebra-fication of Geometry...
...or geometri-fication of algebra! Descartes brought geometry and algebra together for the first time.
The fledgling field of algebra ("al-jabr" from the Arabic, "reunion of broken parts" ) was slowly creeping into European circles...along with the decimal point (Galileo had neither), and solutions of some kinds of polynomial equations were appearing. The notation was clumsy.
So geometry held on as king of mathematics. What Descartes did was link the solutions of geometry problems---which would have been done with rule-obsessive construction of proofs---to solutions using symbols. He did this work in a small book called *Le Géométrie* (*The Geometry*), which he published in 1637, the same year he published his philosophical blockbuster, *Discourse on Method*. In it he instituted a number of conventions which we use today. For example, he reserved the letters of the beginning of the alphabet $a, b, c,...$ for things that are constants or which represent fixed lines. An important strategic approach was to assume that the solution of a mathematical problem may be unknown, but can still be found, and he reserved the last letters of the alphabet $x, y, z...$ to stand for unknown quantities—variables. He further introduced the compact notation of exponents to describe how many times a constant or a variable is multiplied by itself, $x^2$ for example.
The early translators of al-jabr to, um, algebra considered equations in two unknown variables like $y = \text{some combination of } x$'s to be unsolvable. But Descartes linked one variable, say $y$ to the other as points on a curve that related them through an algebraic equation---what came to be called a **function**. He called one of those variable's domain the abscissa and the other, the ordinate. The use of perpendicular axes, which we call $x$ and $y$, stems from Descartes’ inspiration which is why they’re called *Cartesian Coordinates*.
Mathematicians picked up on these ideas and extended them into the directions that we know and love. One of those was John Wallis (1616-1703), the most important Cambridge influence on Isaac Newton.
### Descartes’ Philosophy: New Knowledge Just By Thinking?
The rigor of the mathematical deductive method stayed with him and became a new kind of philosophy that he called "analytic." Famously, he convinced himself that he had deduced a method to truth: whatever cannot be logically doubted, is true. The clue was that when you mentally and relentlessly doubted something and can’t go any further, then that idea has become "clear and distinct." True, for him. Using this method, he decided that this demonstrated that his mind exists and that he, a thinker, is thinking these things and therefore *he* exists.
So by using a mathematical-like deductive path, he believed that he had made an important discovery---a proof of his existence. This is his famous bumper sticker conclusion called forever "the *cogito*": *Cogito ergo sum,* I think, therefore, I am. But that's not what he wrote in *Meditations on First Philosophy*. This is closer: "So after considering everything very thoroughly, I must finally conclude that this proposition, I am, I exist, is necessarily true whenever it is put forward by me or conceived in my mind." Big bumper. But you know how legends go.
```{aside}
Descartes believed that when an idea was incapable of being doubted, then you can believe it. But the Bible and pretty much all that Aristotle wrote and with it the post-Thomas Aquinas Church teachings were off-limits to doubt. In fact those sources were themselves the only authority used to determine truth and falsity. Descartes pretty much changed that in philosophy.
That's why he felt safer in the Netherlands. Not only was he questioning Authority, he'd been inspired by Galileo’s telescopic discoveries and became a committed Copernican. In 1633 he was completely spooked by the Galileo's troubles with the Inquisition and so he stayed put. Eventually, the Pope did ban his writings, 12 years after he died.
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This is the philosophy of **Rationalism** of which he is the king---the discovery of knowledge through pure thought. Rationalism has been in direct philosophical conflict with the philosophy of **Empiricism**--- and as you’ll see, often physics is caught in the middle. Rationalism is in the spirit of Plato, but unlike Descartes, the Greek gave up on the sensible world as simply a bad copy of the Real World, which is one of Ideas...”out there” somewhere. By contrast, by asserting that *mind* and *matter* were both existent realms, Descartes decided that one could understand the universe by blending thinking (mind) with observing (body).
We physicists take some inspiration through Descartes’ approach. Theoretical physicists are often motivated to gain knowledge through thought, always deploying mathematics---so maybe thought and paper. Experimental physicists sometimes claim that knowledge can only be obtained through observation (and in modern form, experiment). Most of us are of the latter devotion, but can sometimes be amazed at how often smart physicists by just thinking can lead to new knowledge of the world. We’ll meet many of these folks. It’s sometimes a strange way to make a living.
```{aside}
One of Descartes' applications of his Rationalism was assigning the postulates in order to do his physics---a mechanical cause for all phenomena. So he populated the universe with invisible balls moving in vortices (to move the planets) and little left-handed screw-type objects to cause magnetism, to name two. Science without such prior causes was unacceptable to him and led to an eventual dispute between the "Cartesian" French and continental scientists and the "Newtonian" British scientists...since Newton decided that such "hypotheses" were unnecessary. We'll visit this in a bit.
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After a public dispute---even in the Netherlands---Descartes began to imagine that his time among the Dutch was coming to a close. Queen Christina of Sweden was an admirer and an intellectual, and she invited Descartes to Stockholm to work in her court and to instruct her. After multiple refusals, not being a monarch to whom "no" is an easy answer, she sent a ship to Amsterdam to pick him up. He eventually accepted the position which was the beginning of his end.
The Queen required his presence at 4 AM for lessons. This, from the fellow who had spent every morning of his life in bed until noon! He caught a serious respiratory infection and died on February 11th, 1650 at the age of only 53.
We moderns owe an enormous debt to this soldier-philosopher-mathematician. Both for what he said that was useful and for what he said that was nonsense, but which stimulated a productive reaction. I think that there is a direct line from every QS&BB lesson that goes right back to René Descartes.